Density of the Level Sets of the Metric Mean Dimension for Homeomorphisms

Abstract

Let N be an n-dimensional compact riemannian manifold, with \(n\ge 2\) . In this paper, we prove that for any \(\alpha \in [0,n]\) , the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to \(\alpha \) is dense in \(\text {Hom}(N)\) . More generally, given \(\alpha ,\beta \in [0,n]\) , with \(\alpha \le \beta \) , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to \(\alpha \) and upper metric mean dimension equal to \(\beta \) is dense in \(\text {Hom}(N)\) . Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in \(\text {Hom}(N)\) .